ORTHOGONAL POLYNOMIALS DEFINED BY SELF-SIMILAR MEASURES WITH OVERLAPS

MEASURES WITH OVERLAPS

SZE-MAN NGAI, WEI TANG, ANH TRAN, AND SHUAI YUAN

Abstract. We study orthogonal polynomials with respect to self-similar measures, focusing on the class of infinite Bernoulli convolutions, which are defined by iterated function systems with overlaps, especially those defined by the Pisot, Garsia, and Salem numbers. By using an algorithm of Mantica, we obtain graphs of the coeffi- cients of the 3-term recursion relation defining the orthogonal polynomials. We use these graphs to predict whether the singular infinite Bernoulli convolutions belong to the Nevai class. Based on our numerical results, we conjecture that all infinite Bernoulli Convolutions with contraction ratios greater than or equal to 1/2 belong to Nevai’s class, regardless of the probability weights assigned to the self-similar measures.

1. Introduction

Orthogonal polynomials with respect to fractal measures have been studied by authors including Mantica [9,10], Heilmanet al [7], Kr¨uger and Simon [8], and Bandt and Pe˜na [1]. In [7], the self-similar measures studied satisfy the open set condition.

As the classical theory [6, 22] to compute the orthogonal polynomials can be applied to a large class of general measures and algorithms formulated by Mantica to compute the coefficients of the 3-term recurrence relation (see (2.2)) can be applied to general self-similar measures, we use them to study self-similar measures defined by iterated function systems with overlaps. This paper can be consider a continuation of the explorations in [7].

Let µbe a positive measure on [0,1]. We say that a measure µ is in Nevai’s class (see [12, 14, 18]) if the recurrence coefficients A_n and r_n in (2.2) satisfy

n→∞lim A_n= 1

2 and lim

n→∞r_n= 1 4.

Date: June 1, 2019.

2010Mathematics Subject Classification. Primary: 28A80, 42C05; Secondary: 33C45, 28A25.

Key words and phrases. Orthogonal polynomial, self-similar measure with overlaps, Nevai class.

The first two authors are supported in part by the National Natural Science Foundation of China, Grants 11771136 and 11271122, and Construct Program of the Key Discipline in Hunan Province.

The first author is also supported in part by the Hunan Province Hundred Talents Program, the Center of Mathematical Sciences and Applications (CMSA) of Harvard University. SN, AT, and SY are also supported in part by a Faculty Research Scholarly Pursuit Award from Georgia Southern University.

1

Rahmanov [19, 20] proved that if µ⁰ > 0 for Lebesgue a.e. on [0,1], then µ belongs to Nevai’s class. On the other hand, Lubinsky [13] constructed singular continuous measures that belong to Nevai’s class. These results seem to suggest the relationship between absolute continuity being in Nevai’s class, and motivated our study of infinite Bernoulli convolutions with overlaps. They are self-similar measures defined by an iterated function system (IFS) with two similitudes{S₁, S₂}together with probability weights w₁ =w₂ = 1/2 as follows:

µ= 1

2µ◦S₁⁻¹+ 1

2µ◦S₂⁻¹, where

S₁(x) = ρx, S₂(x) = ρx+ (1−ρ), 1/2≤ρ <1.

(1.1) We are particularly interested in several families of algebraic integers. If ρ⁻¹ is a Pisot number (i.e., an algebraic integer >1 whose algebraic conjugates lie inside the unit circle), then µ is singular [3]. This is the only class of numbers for which µ is known to be singular. Wintner [23] proved that if ρ = 1/√ⁿ

2 for some n ∈ N, then µ is absolutely continuous. Garsia [5] constructed a class of algebraic integers for which µ is absolutely continuous. These are the only numbers for which the self- similar measures are explicitly known to be absolute continuous, and by Rahmanov’s theorem, they are in Nevai’s class. Solomyak [21] proved that for Lebesgue a.e.

ρ∈(1/2,1), µis absolutely continuous. Salem numbers are algebraic integers greater than 1 whose algebraic conjugates have moduli less than or equal to 1, with at least one of them being 1. They are of interest in dynamical systems and number theory (see [2, 15] and references therein) as well as fractals (see, e.g., [4]). Although the definition of a Salem number is close to that of a Pisot numbers, Salem numbers, as well as the associated self-similar measures, are much less understood. It is not known whether the self-similar measure associated with a Salem number is absolutely continuous or singular.

We also study weighted infinite Bernoulli convolutions, namely,

µ=w₁µ◦S₁⁻¹+w₂µ◦S₂⁻¹, wherew₁ >0, w₂ >0, w₁+w₂ = 1,

S₁(x) =ρx, S₂(x) =ρx+ (1−ρ), and 1/2≤ρ <1. (1.2) As the weights become more biased, the corresponding measure µ tend to be more singular (in the sense that the Hausdorff dimension ofµbecomes smaller). However, our numerical experiments suggest, rather unexpectedly, that µ continues to belong to Nevai’s class. Unfortunately, we are not able to prove this.

This paper is organized as follows. In Section 2, we establish some properties of the recurrence coefficients for general measures. In Section 3, we graph the orthogonal polynomials defined by the infinite Bernoulli convolution associated with the gold- en ratio and the 3-fold convolution of the Cantor measure, both being self-similar measures with overlaps that have been studied extensively. In Section 4 we derive

explicit formulas for computing the recurrence coefficients by using both Mantica’s and Chebyshev’s algorithms. Numerical solutions are displayed in Section 5. Finally, we state some open problems and conjectures in Section 6.

2. 3-term recursive relation and symmetric measures

Letµbe a positive measure on R. LetPbe the vector space of all real polynomials and Pⁿ be the subspace of polynomials of degree ≤ n. Define an inner product and norm on P respectively as

hu, vi_µ:=

Z

R

uv dµ, kuk_µ :=

q

hu, ui_µ.

The inner product is said to be positive definite on P (resp. Pn) if kuk_µ > 0 for all u ∈ P (resp. Pn) that is not identically 0. The orthogonal polynomials {P_n(x)}n≥0

with respect to µare polynomials of the form

Pn(x) =cnxⁿ+· · ·+c1x+c0 with cn >0 (2.1) that satisfy hP_i, P_ji_µ = δ_ij for all i, j = 0,1,2, . . ., where δ_ij is the Kronecker delta.

{P_i(x)}ⁿ_i=0form a basis ofPn. The existence and uniqueness of orthogonal polynomials {P_n(x)}n≥0 with respect toµare well-known (see [6, Theorem 1.6]). The orthogonal polynomials {P_n(x)}n≥0 satisfy a 3-term recurrence relation (see [6, Theorem 1.29])

xP_n(x) =r_n+1P_n+1(x) +A_nP_n(x) +r_nPn−1(x) for n = 0,1,2. . . , (2.2) where P−1(x) = 0, P₀(x) = 1, and r₀ = 1. Orthogonality implies that for all n ≥0,

r_n+1 =hxP_n+1, P_ni_µ and A_n=hxP_n, P_ni_µ.

Monic real polynomials Pe_n(x) = xⁿ +· · · , n = 0,1,2, . . . , are called the monic orthogonal polynomials with respect to the measure µ if hP_i, P_ji_µ = 0 for all i6=j ∈ {0,1,2, . . .} and kPe_ik_µ > 0 for all i = 0,1,2, . . .. Note that P_n(x) = Pe_n(x)/kPe_nk_µ. Combining [6, Theorem 1.29] and (2.2), we see that monic orthogonal polynomials {Pen(x)}n≥0 satisfy the following 3-term recurrence relation

xPe_n(x) =Pe_n+1(x) +A_nPe_n(x) +r_n²Pen−1(x) for n = 0,1,2. . . , (2.3) where Pe−1(x) = 0 and Pe₀(x) = 1.

Let x₀ ∈R and T(x) := 2x₀ −x for x∈ R. Note that T =T⁻¹. We say that µis symmetric with respect to x=x₀ if µ(E) =µ◦T⁻¹(E) for all µ-measurableE ⊆R. Proposition 2.1. Let µ be a positive measure on R and let {P_n(x)} be orthogonal polynomials with respect to µ as in 2.1. Assume that µ is symmetric with respect to x=x₀ for some x₀ ∈R and let T(x) := 2x₀−x. Then for each n≥0,

(a) P_n(x) = (−1)ⁿP_n(T(x)) for all x∈R;

(b) A_n=x₀.

Proof. (a) For eachn ≥0, define ˆP_n(x) := (−1)ⁿP_n(T(x)) for x∈R. It follows from the definitions of T(x) and P_n(x) that ˆP_n(x) =c_nxⁿ+· · ·. Moreover, for 0≤k 6=`, we have

hPˆ_k,Pˆ<sub>`</sub>i<sub>µ</sub> = (−1)<sup>k+`</sup>

Z

R

P_k(T(x))P_`(T(x))dµ(x)

= (−1)^k+`

Z

T(R)

P_k(y)P_`(y)dµ◦T⁻¹(y)

= (−1)^k+`

Z

R

P_k(y)P<sub>`</sub>(y)dµ(y) = (−1)<sup>k+`</sup>hP_k, P_{`</sub>i<sub>µ</sub> =δ<sub>k`}.

(2.4)

Thus{Pˆ_n(x)}n≥0 is also a sequence of orthogonal polynomials with respect to µ. By the uniqueness of orthogonal polynomials, P_n(x) = ˆP_n(x), i.e., (a) holds.

(b) CombiningT =T⁻¹, µ=µ◦T⁻¹, and (a), we have A_n=

Z

R

xP_n²(x)dµ(x) = Z

T(R)

(T⁻¹(y))P_n²(T⁻¹(y))dµ◦T⁻¹(y)

= Z

R

(2x₀−y)P_n²(T(y))dµ(y) = Z

R

(2x₀−y)P_n²(y)dµ(y)

= 2x0−An for n ≥0.

(2.5)

ThusAn=x0 for all n≥0.

3. Graphing orthogonal polynomials with respect to self-similar measures

Letµ be the self-similar measure defined by an IFS{S_i}^N_i=1 on R of the form S_i(x) =ρ_ix+b_i, i= 1, . . . , N, (3.1) together with a probability weight {w_i}^N_i=1. That is, µ is the unique probability measure satisfying the following self-similar identity

µ=

N

X

i=1

w_iµ◦S_i⁻¹. (3.2)

Thus for any continuous function f onR, Z

K

f dµ=

N

X

i=1

w_i Z

K

f ◦S_idµ, (3.3)

whereK := supp(µ) is called theself-similar set defined by {Si}^N_i=1. It is known that if N ≥2, then µis continuous. For n ≥ 0, let m_n be the n-th moment with respect

toµ, defined as

m_n=m_n(µ) :=

Z

K

xⁿdµ. (3.4)

Let {Pe_n(x)}n≥0 be the monic orthogonal polynomials with respect to µ. In this section, we will draw the figures of Pe_n(x). Since the monic orthogonal polynomials Pe_k(x) can be expressed in terms of the moments in the following well-known formula (see, e.g., [6]):

Pe_n(x) = 1 m_n

m0 m1 · · · mn

m₁ m₂ · · · m_n+1 ... ... · · · ... mn−1 m_n · · · m2n−1

1 x · · · xⁿ

, (3.5)

we first need to compute the moments {m_i}²ⁿ⁻¹_i=0 with respect toµ.

The following proposition gives a straight-forward derivation of a formula for the moments corresponding to any one-dimensional self-similar measure µ.

Proposition 3.1. Let µ be defined by (3.1) and (3.2), and let m_n be defined as in (3.4). Then

m_n = 1

1−PN i=1w_iρⁿ_i

N

X

i=1

w_i

n−1

X

k=0

n k

ρ^k_ib^n−k_i m_k.

Proof. Lettingf(x) =xⁿ in (3.3) yields m_n =

N

X

i=1

w_i Z

K

(ρ_ix+b_i)ⁿdµ=

N

X

i=1

w_i Z

K

ρⁿ_ixⁿ+

n−1

X

k=0

n k

ρ^k_ib^n−k_i x^k

dµ

=X^N

i=1

w_iρⁿ_i m_n+

N

X

i=1

w_i

n−1

X

k=0

n k

ρ^k_ib^n−k_i m_k,

which gives the desired formula.

3.1. Infinite Bernoulli convolution associated with the golden ratio. The infinite Bernoulli convolution associated with golden ratioµis the self-similar measure on [0,1] defined by the IFS

S1(x) =ρx, S2(x) =ρx+ (1−ρ), ρ= (√

5−1)/2≈0.618033988. . . , (3.6) together with probability weights w₁ =w₂ = 1/2. That is,

µ= 1

2µ◦S₁⁻¹+1

2µ◦S₂⁻¹. (3.7)

It is known that µ is singular (see [3]) with supp(µ) = [0,1]. By using Proposition 3.1, we have

m_n= 1 2

ρ²ⁿ (1−ρⁿ)

n−1

X

i=0

n i

ρ⁻ⁱm_i. (3.8)

Now the point is how to compute the moments. Using formula (3.8), would takes too much time to compute m_n when n > 18. The following are some equalities which may be used to reduce computing time. We first introduce the Lucas and Fibonacci sequences, which are closely related to the golden ratio ρ. The Lucas sequence {L_n}n≥0 is given by the recurrence relation:

L₀ = 2, L₁ = 1, L_n=Ln−1 +Ln−2, n∈Z, (3.9) while the Fibonacci sequence {F_n}n≥0 is given by

F0 = 0, F1 = 1, Fn=Fn−1+Fn−2, n ∈Z. (3.10) Letφ :=ρ⁻¹. We note that for n ≥0,

Ln =φⁿ+ (1−φ)ⁿ and Fn= φⁿ−(1−φ)ⁿ

√5 . Thus

φⁿ= (L_n+√

5F_n)/2 for n≥0. (3.11)

It follows that ρ²ⁿ

1−ρⁿ = 1

ρ⁻²ⁿ−ρ⁻ⁿ = 1

φ²ⁿ−φⁿ = 2

(L_2n−L_n) +√

5(F_2n−F_n). (3.12) Combining (3.8), (3.11) and (3.12), we have

m_n = 1

2 · 1

(L2n−Ln) +√

5(F2n−Fn)

n−1

X

i=0

n i

(L_i+√

5F_i)m_i.

Monic orthogonal polynomials Pe₁(x), . . . ,Pe₅(x) corresponding to the golden ratio are plotted in Figure 1(a).

3.2. Three-fold convolution of the Cantor measure. The 3-fold convolution of the Cantor measure is the self-similar measure defined by the IFS

S_i(x) = 1 3x+ 2

3(i−1), i= 1,2,3,4, together with the probability weight {1/8,3/8,3/8,1/8},i.e.,

µ= 1

8µ◦S₁⁻¹ +3

8µ◦S₂⁻¹+3

8µ◦S₃⁻¹+ 1

8µ◦S₄⁻¹.

0.0 0.2 0.4 0.6 0.8 1.0 -0.4

-0.2 0.0 0.2 0.4

(a)

0.0 0.5 1.0 1.5 2.0 2.5 3.0

-2 -1 0 1 2

(b)

Figure 1. Monic orthogonal polynomialsPe₁(x), . . . ,Pe₅(x) correspond- ing to (a) the infinite Bernoulli convolution associated with the golden ratio, and (b) the three-fold convolution of the Cantor measure.

0 3

S₁

S_{2 C}

C CW

S_{3 A}

A A U

S₄

1 2

0 3

Figure 2. Figure showing the IFS defining the 3-fold convolution.

(See Figure 3.2.) We note that µ = µ^∗3_c := µ_c∗µ_c ∗µ_c, where µ_c is the standard Cantor measure.

Proposition 3.1 implies that for the 3-fold convolution of the Cantor measure, the moments mn satisfy

m_n = 1 8(3ⁿ−1)

n−1

X

k=0

n k

(3·2^n−k+ 3·4^n−k+ 6^n−k)m_k.

Monic orthogonal polynomials Pe₁(x), . . . ,Pe₅(x) corresponding to the 3-fold convolu- tion are plotted in Figure 1(b).

4. Algorithms for computing A_n and r_n

In this section, we introduce two methods, namely Mantica’s algorithm [9] and modified Chebyshev’s algorithm [6], to compute the A_n’s andr_n’s in (2.2).

4.1. Mantica’s algorithm. In this subsection, we use a method of Mantica [9] to compute the A_n’s and r_n’s in (2.2). We first describe the method for an IFS {S_i}^N_i=1 onR, where

S_i(x) =ρ_ix+b_i, i= 1, . . . , N,

and let µbe the self-similar measure defined by the IFS together with a set of prob- ability weights {w_i}^N_i=1.

Let P_n(x) be orthogonal polynomials with respect to µ. For any n ≥ 0 and i ∈ {1, . . . , N}, and 0≤`≤n, let Γ<sup>n</sup><sub>i,`</sub> be defined by the relation:

P_n(S_i(x)) =

n

X

`=0

Γⁿ_{i,`</sub>P<sub>`}(x). (4.1)

According to [9, Lemma 1], the coefficients Γⁿ_i,` can be determined recursively from {ρ_i}^N_i=1, {b_i}^N_i=1, {A_k}ⁿ⁻¹_k=0, and {r_k}ⁿ_k=0. We state the precise result below.

Proposition 4.1. Let (Γⁿ_i,`)be the the coefficients in (4.1). For each i∈ {1, . . . , N}, we have

(a) Γ¹_i,0 = ρ_iA₀Γ⁰_i,0+b_iΓ⁰_i,0−A₀Γ⁰_i,0

/r₁ and Γ¹_i,1 =ρ_iΓ⁰_i,0. (b)

Γ²_i,0 = ρ_ir₁Γ¹_i,1+ρ_iA₀Γ¹_i,0+b_iΓ¹_i,0−A₁Γ¹_i,0 −r₁Γ⁰_i,0 /r₂ Γ²_i,1 = ρir1Γ¹_i,0+ρiA1Γ¹_i,1+biΓ¹_i,1−A1Γ¹_i,1

/r2

Γ²_i,2 =ρ_iΓ¹_i,1.

(4.2)

(c) for n≥3 and 1≤`≤n−2,

Γⁿ_i,0 = ρ_iA₀Γⁿ⁻¹_i,0 +ρ_ir₁Γⁿ⁻¹_i,1 +b_iΓⁿ⁻¹_i,0 −An−1Γⁿ⁻¹_i,0 −rn−1Γⁿ⁻²_i,0 /r_n;

Γⁿ_{i,`</sub>= ρ<sub>i</sub>r<sub>`}Γⁿ⁻¹_{i,`−1</sub>+ρ<sub>i</sub>A<sub>`}Γⁿ⁻¹_{i,`</sub> +ρ<sub>i</sub>r<sub>`+1}Γⁿ⁻¹_{i,`+1</sub>+b<sub>i</sub>Γ<sup>n−1</sup><sub>i,`} −An−1Γⁿ⁻¹_{i,`</sub> −rn−1Γ<sup>n−2</sup><sub>i,`}

/r_n; Γⁿ_i,n−1 = ρ_irn−1Γⁿ⁻¹_i,n−2+ρ_iAn−1Γⁿ⁻¹_i,n−1 +b_iΓⁿ⁻¹_i,n−1−An−1Γⁿ⁻¹_i,n−1

/r_n; Γⁿ_i,n=ρ_iΓⁿ⁻¹_i,n−1.

Proof. We first note thatP−1(x) = 0. Fix any i∈ {1, . . . , N}.

(a) Using 3-term recurrence relation (2.2), we have

xP₀(x) =r₁P₁(x) +A₀P₀(x) +r₀P−1(x) =r₁P₁(x) +A₀P₀(x). (4.3) Substituting xby Si(x) = ρix+bi, we have

(ρ_ix+b_i)P₀(S_i(x)) =r₁P₁(S_i(x)) +A₀P₀(S_i(x)).

It follows from (4.1) that (ρ_ix+b_i)Γ⁰_i,0P₀(x) =r₁ Γ¹_i,0P₀(x) + Γ¹_i,1P₁(x)

+A₀Γ⁰_i,0P₀(x), i.e.,

ρ_iΓ⁰_i,0xP₀(x) = r₁Γ¹_i,0+A₀Γ⁰_i,0−b_iΓ⁰_i,0

P₀(x) +r₁Γ¹_i,1P₁(x). (4.4) Combining (4.4) with (4.3), we have

ρ_iΓ⁰_i,0 r₁P₁(x) +A₀P₀(x)

= r₁Γ¹_i,0+A₀Γ⁰_i,0−b_iΓ⁰_i,0

P₀(x) +r₁Γ¹_i,1P₁(x). (4.5) Thusρ_ir₁Γ⁰_i,0 =r₁Γ¹_i,1 and ρ_iA₀Γ⁰_i,0 =r₁Γ¹_i,0+A₀Γ⁰_i,0−b_iΓ⁰_i,0. Consequently,

Γ¹_i,1 =ρ_iΓ⁰_i,0, Γ¹_i,0 = ρ_iA₀Γ⁰_i,0+b_iΓ⁰_i,0−A₀Γ⁰_i,0 /r₁. (b) Similarly, using 3-term recurrence relation (2.2), we have

xP₁(x) = r₂P₂(x) +A₁P₁(x) +r₁P₀(x). (4.6) Substituting xby S_i(x) = ρ_ix+b_i, we have

(ρ_ix+b_i)P₁(S_i(x)) =r₂P₂(S_i(x)) +A₁P₁(S_i(x)) +r₁P₀(S_i(x)). (4.7) Using (4.1), (4.3), and (4.6), we have the left side of (4.7)

(ρ_ix+b_i)P₁(S_i(x)) = (ρ_ix+b_i) Γ¹_i,0P₀(x) + Γ¹_i,1P₁(x)

=ρ_iΓ¹_i,0xP₀(x) +ρ_iΓ¹_i,1xP₁(x) +b_iΓ¹_i,0P₀(x) +b_iΓ¹_i,1P₁(x)

=ρ_iΓ¹_i,0 r₁P₁(x) +A₀P₀(x)

+ρ_iΓ¹_i,1 r₂P₂(x) +A₁P₁(x) +r₁P₀(x) +biΓ¹_i,0P0(x) +biΓ¹_i,1P1(x)

=ρ_ir₂Γ¹_i,1P₂(x) + ρ_ir₁Γ¹_i,0+ρ_iA₁Γ¹_i,1+b_iΓ¹_i,1 P₁(x) + ρ_iA₀Γ¹_i,0+ρ_ir₁Γ¹_i,1+b_iΓ¹_i,0

P₀(x).

(4.8) On the other hand, (4.1) implies that the right side of (4.7)

r₂P₂(S_i(x)) +A₁P₁(S_i(x)) +r₁P₀(S_i(x))

=r₂ Γ²_i,0P₀(x) + Γ²_i,1P₁(x) + Γ²_i,2P₂(x)

+A₁(Γ¹_i,0P₀(x) + Γ¹_i,1P₁(x) +r₁Γ⁰_i,0P₀(x)

=r2Γ²_i,2P2(x) + r2Γ²_i,0+A1Γ¹_i,0+r1Γ⁰_i,0

P0(x) + r2Γ²_i,1+A1Γ¹_i,1 P1(x).

(4.9)

Combining (4.7), (4.7), and (4.7), we obtain

Γ²_i,0 = ρir1Γ¹_i,1+ρiA0Γ¹_i,0+biΓ¹_i,0−A1Γ¹_i,0 −r1Γ⁰_i,0 /r2

Γ²_i,1 = ρ_ir₁Γ¹_i,0+ρ_iA₁Γ¹_i,1+b_iΓ¹_i,1−A₁Γ¹_i,1 /r₂ Γ²_i,2 =ρ_iΓ¹_i,1.

(4.10)

(c) Fix any n≥3. Replacing x bySi(x) =ρix+bi in (2.2) gives

(ρ_ix+b_i−A_n−1)P_n−1(S_i(x)) =r_nP_n(S_i(x)) +r_n−1P_n−2(S_i(x)). (4.11)

Using (4.1), we can write this as (ρ_ix+b_i−An−1)

n−1

X

`=0

Γⁿ⁻¹_{i,`</sub> P<sub>`}(x) = r_n

n

X

`=0

Γⁿ_{i,`</sub>P<sub>`}(x) +rn−1 n−2

X

`=0

Γⁿ⁻²_{i,`</sub> P<sub>`}(x)

=

n−2

X

`=0

r_nΓⁿ_{i,`</sub>+rn−1Γ<sup>n−2</sup><sub>i,`} )P_`(x) +r_nΓⁿ_i,n−1Pn−1(x) +r_nΓⁿ_i,nP_n(x).

(4.12)

Moreover, (2.2) implies (ρ_ix+b_i−An−1)

n−1

X

`=0

Γⁿ⁻¹_{i,`</sub> P<sub>`}(x)

=ρ_i

n−1

X

`=0

Γⁿ⁻¹_{i,`</sub> xP<sub>`}(x) + (b_i−An−1)

n−1

X

`=0

Γⁿ⁻¹_{i,`</sub> P<sub>`}(x)

=ρ_i

n−1

X

`=0

Γⁿ⁻¹_{i,`</sub> r<sub>`+1}P_{`+1</sub>(x) +A<sub>`}P_{`</sub>(x) +r<sub>`}P`−1(x)

+ (b_i−An−1)

n−1

X

`=0

Γⁿ⁻¹_{i,`</sub> P<sub>`}(x)

= ρ_iA₀Γⁿ⁻¹_i,0 +ρ_ir₁Γⁿ⁻¹_i,1 +b_iΓⁿ⁻¹_i,0 −An−1Γⁿ⁻¹_i,0 P₀(x) +

n−2

X

`=1

ρ_ir_{`</sub>Γ<sup>n−1</sup><sub>i,`−1}+ρ_iA_{`</sub>Γ<sup>n−1</sup><sub>i,`} +ρ_ir_{`+1</sub>Γ<sup>n−1</sup><sub>i,`+1}+b_iΓⁿ⁻¹_{i,`</sub> −A<sub>n−1</sub>Γ<sup>n−1</sup><sub>i,`}

P_`(x) + ρ_ir_n−1Γⁿ⁻¹_i,n−2+ρ_iA_n−1Γⁿ⁻¹_i,n−1+b_iΓⁿ⁻¹_i,n−1−A_n−1Γⁿ⁻¹_i,n−1

P_n−1(x) +ρ_ir_nΓⁿ⁻¹_i,n−1P_n(x).

(4.13) Equating (4.12) and (4.12) gives, for 1≤`≤n−2,

Γⁿ_i,0 = ρ_iA₀Γⁿ⁻¹_i,0 +ρ_ir₁Γⁿ⁻¹_i,1 +b_iΓⁿ⁻¹_i,0 −A_n−1Γⁿ⁻¹_i,0 −r_n−1Γⁿ⁻²_i,0 /r_n;

Γⁿ<sub>i,`</sub>= ρir`Γⁿ⁻¹<sub>i,`−1</sub>+ρiA`Γⁿ⁻¹<sub>i,`</sub> +ρir`+1Γⁿ⁻¹_{i,`+1</sub>+biΓ<sup>n−1</sup><sub>i,`} −An−1Γⁿ⁻¹_{i,`</sub> −rn−1Γ<sup>n−2</sup><sub>i,`}

/rn; Γⁿ_i,n−1 = ρ_irn−1Γⁿ⁻¹_i,n−2+ρ_iAn−1Γⁿ⁻¹_i,n−1 +b_iΓⁿ⁻¹_i,n−1−An−1Γⁿ⁻¹_i,n−1

/r_n; Γⁿ_i,n=ρ_iΓⁿ⁻¹_i,n−1.

The proof is complete.

LetPe_n(x) =r_nP_n(x). Hence, we can write a second decomposition Pe_n(S_i(x)) = eΓⁿ_i,`Pe_n(x) +

n−1

X

`=0

Γeⁿ_{i,`</sub>P<sub>`}(x). (4.14) It is easy to see that Γeⁿ_{i,`</sub> = r<sub>n</sub>Γ<sup>n</sup><sub>i,`} for 0 ≤ ` ≤ n− 1 and eΓⁿ_i,n = Γⁿ_i,n. Thus the coefficients Γⁿ_i,n can be computed recursively on the basis of the knowledge of only (r_k)ⁿ⁻¹_k=1 and (A_k)ⁿ⁻¹_k=1.

The following proposition gives formulas for An and rn. The proof is given in [9, Lemmas 2 and 3].

Proposition 4.2. For n≥0, (a) r²_n=

PN

i=1w_i(B_i+C_i)

/ 1−PN

i=1w_iρ_ieΓⁿ_i,nΓⁿ⁻¹_i,n−1

, where

B_i =

n−1

X

`=0

(b_i+ρ_iA_{`</sub>)eΓ<sup>n</sup><sub>i,`}Γⁿ⁻¹_i,` , and C_i =ρ_i

n−2

X

`=0

r_{`+1</sub> eΓ<sup>n</sup><sub>i,`}Γⁿ⁻¹_{i,`+1</sub>+Γe<sup>n</sup><sub>i,`+1}Γⁿ⁻¹_i,`

. (4.15) (b) A_n=PN

i=1w_i Pn−1

`=0(Γ<sup>n</sup><sub>i,`</sub>)²(b_i+ρ_iA_`)+2ρ_iPn−1

`=0 r<sub>`+1</sub>Γⁿ_{i,`</sub>Γ<sup>n</sup><sub>i,`+1}+b_i Γⁿ_i,n2 /(1−

PN

i=1w_iρ_i Γⁿ_i,n2

).

4.2. Modified Chebyshev’s algorithm. Let µ be a positive measure on R with compact or infinite support, for which all moments m_n, n = 0,1, . . ., exist and are finite. Let {Pe_n(x)}n≥0 be the monic orthogonal polynomials with respect to µ. We can also use the Modified Chebyshev algorithm [6] to compute the A_n’s and r_n’s in (2.2), as follows. Let {π_n(x)}n≥0 denote a system of monic polynomials satisfying a recurrence relation

xπ_n(x) =π_n+1(x) +α_nπ_n(x) +β_nπn−1(x), n = 0,1,2, . . . ,

π−1(x) = 0, π₀(x) = 1, (4.16)

where α_n ∈ R and β_n ≥ 0 are assumed known. In the case α_n =β_n = 0, it reduces to Chebyshev’s original algorithm. We then call

v_n =v_n(µ) :=

Z

R

π_n(x)dµ(x), n= 0,1,2, . . . , (4.17) the modified moments of the measure µ relative to the polynomial system {π_n(x)}.

Combining (2.3) and (4.16), we see that ifα_n=A_n and β_n =r_n², thenπ_n(x) = Pe_n(x) and the modified moments reduce to the ordinary moments (3.4).

To describe the algorithm, we introduce mixed moments, which are defined for k, `≥ −1, as follows

σ_k,` :=

Z

R

Pe_k(x)π<sub>`</sub>(x)dµ with σ−1,` := 0. (4.18) One obtains the first n coefficients αk, βk, k = 0,1, . . . , n−1, from the first 2n modified moments vi, i = 0,1, . . . ,2n− 1, by the following modified Chebyshev’s algorithm:

Initialization:

A₀ =α₀+v₁/v₀, r₀ =√

v₀,

σ−1,` = 0, `= 1,2, . . . ,2n−2.

σ_{0,`</sub>=v<sub>`}, `= 0,1,2, . . . ,2n−1.

(4.19)

Continuation: For k= 1,2, . . . , n−1,

σ_{k,`</sub> =σ<sub>k−1,`+1}−(A_k−1−α_{`</sub>)σ<sub>k−1,`}−r_k−1² σ_{k−2,`</sub>+β<sub>`}σ_k−1,`−1,

` =k, k+ 1, . . . ,2n−k−1, A_k =α_k+ σ_k,k+1

σ_k,k − σk−1,k

σ_k−1,k−1, r_k =

r σ_k,k σk−1,k−1

.

(4.20)

The algorithm requires{v`}<sup>2n−1</sup><sub>`=0</sub> and {αk, βk}²ⁿ⁻²_k=0 as input, and produces{Ak, rk}ⁿ⁻¹_k=0. The complexity in terms of arithmetic operations is O(n²). We apply the Chebyshev algorithm to infinite Bernoulli convolutions with overlaps. The results are the same as those obtained by Mantica’s algorithm.

5. Numerical results for infinite Bernoulli convolutions

In this section, we display numerical results for the recurrence coefficients An and rn in (2.2) for the symmetric and weighted infinite Bernoulli convolutions in (1.1) and (1.2). The value of ρ in (1.1) and (1.2) are taken from the following Table 1 through Table 4. For symmetric Bernoulli convolutions, we have A_n = 1/2 for all n = 0,1,2, . . . (see Proposition 2.1), and so in this case, we only display numerical results for r_n. For the weighted infinite Bernoulli convolutions in (1.2) with weights w₁ = 1/4 and w₂ = 3/4, we show numerical results for both A_n and r_n.

We also remark that for the symmetric Bernoulli convolutions, A_n= 1/2 is known, and so we are able to computernquite efficiently fornup to 10000. For the weighted Bernoulli convolutions, computations ofAnandrn take much more machine time and memory and so we only show results up to n= 1000.

Label ρ⁻¹ Appro. valueρ Label ρ⁻¹ Appro. valueρ (a) √²

2 0.7071067811 (d) √⁸

2 0.9170040432 (b) √⁴

2 0.8408964152 (e) ²⁰√

2 0.9659363289 (c) √⁶

2 0.8908987181 (h) ¹⁰⁰√

2 0.9930924954

Table 1. The sequence √ⁿ 2.

Label Minimal polynomial Appro. value ofρ⁻¹ ρ (a) x³−2x−2 1.7692923542 0.5651977173 (b) x³−x²−2 1.6956207695 0.5897545123 (c) x³−2x²+ 2x−2 1.5436890126 0.6477988712 (d) x³−x²+x−2 1.3532099641 0.7389836215 (e) x⁵−x²−2 1.2980299423 0.7703982530 (f) x³+x²−x−2 1.2055694304 0.8294835409

Table 2. Selected Garsia numbers.

Label Minimal polynomial Appro. value of ρ⁻¹ ρ (a) x⁶−x⁵−x⁴−x³−x²−x−1 1.9835828434 0.5041382583 (b) x³−x²−x−1 1.8392867552 0.5436890126 (c) x³−2x²+x−1 1.7548776662 0.5698402909

(d) x²−x−1 1.6180339887 0.6180339887

(e) x³−x²−1 1.4655712318 0.6823278038

(f) x³−x−1 1.3247179572 0.7548776662

Table 3. (f) is the smallest Pisot number. (d) is the golden ratio. We also include the first few Pisot number from the family xⁿ −xⁿ⁻¹ −

· · · −x−1 = 0. These family of Pisot numbers are decreasing and tend to 1.

Label Minimal polynomial Appro. value of ρ⁻¹ ρ

(a) x⁹−x⁸−x⁷−x⁶−x⁵−x⁴−x³−x²−x+ 1 1.9940041991 0.5015034574 (b) x⁷−x⁶−x⁵−x⁴−x³−x²−x+ 1 1.9748187082 0.5063755957 (c) x⁶−x⁵−x⁴−x³−x²−x+ 1 1.9468562682 0.5136486017 (d) x⁴−x³−x²−x+ 1 1.7220838057 0.5806918319 (e) x¹⁰−x⁸−x⁷+x⁵−x³−x²+ 1 1.2934859531 0.7731046460 (f) x¹⁰+x⁹−x⁷−x⁶−x⁵−x⁴−x³+x+ 1 1.1762808182 0.8501371309

Table 4. Selected Salem numbers.

0 100020003000 400050006000700080009000 10000 0.248

0.2485 0.249 0.2495 0.25 0.2505 0.251 0.2515 0.252

(a)

0 10002000300040005000 6000700080009000 10000 0.248

0.2485 0.249 0.2495 0.25 0.2505 0.251 0.2515 0.252

(b)

0 10002000300040005000600070008000 9000 10000 0.248

0.2485 0.249 0.2495 0.25 0.2505 0.251 0.2515 0.252

(c)

0 1000 20003000400050006000700080009000 10000 0.248

0.2485 0.249 0.2495 0.25 0.2505 0.251 0.2515 0.252

(d)

0 1000200030004000 50006000700080009000 10000 0.248

0.2485 0.249 0.2495 0.25 0.2505 0.251 0.2515 0.252

(e)

0 1000200030004000500060007000 80009000 10000 0.248

0.2485 0.249 0.2495 0.25 0.2505 0.251 0.2515 0.252

(f)

Figure 3. Graphs ofrnfor the infinite Bernoulli convolutions in (1.1), where ρcomes from the sequence {√ⁿ

2} in Table 1.

0 200 400 600 800 1000

0.20 0.22 0.24 0.26 0.28 0.30

(a)

0 200 400 600 800 1000

0.20 0.22 0.24 0.26 0.28 0.30

(b)

0 200 400 600 800 1000

0.20 0.22 0.24 0.26 0.28 0.30

(c)

0 200 400 600 800 1000

0.20 0.22 0.24 0.26 0.28 0.30

(d)

0 200 400 600 800 1000

0.20 0.22 0.24 0.26 0.28 0.30

(e)

0 200 400 600 800 1000

0.20 0.22 0.24 0.26 0.28 0.30

(f)

Figure 4. Graphs ofrnfor the weighted infinite Bernoulli convolutions in (1.2) with weights w1 = 1/4 and w2 = 3/4, whereρ comes from the sequence in {√ⁿ

2}in Table 1.

0 200 400 600 800 1000 0.40

0.45 0.50 0.55 0.60

(a)

0 200 400 600 800 1000

0.40 0.45 0.50 0.55 0.60

(b)

0 200 400 600 800 1000

0.40 0.45 0.50 0.55 0.60

(c)

0 200 400 600 800 1000

0.40 0.45 0.50 0.55 0.60

(d)

0 200 400 600 800 1000

0.40 0.45 0.50 0.55 0.60

(e)

0 200 400 600 800 1000

0.40 0.45 0.50 0.55 0.60

(f)

Figure 5. Graphs of An for the weighted infinite Bernoulli convolu- tions in (1.2) with weightsw₁ = 1/4 andw₂ = 3/4, whereρcomes from the sequence {√ⁿ

2} in Table 1.

0 100020003000 400050006000700080009000 10000 0.248

0.2485 0.249 0.2495 0.25 0.2505 0.251 0.2515 0.252

(a)

0 10002000300040005000 6000700080009000 10000 0.248

0.2485 0.249 0.2495 0.25 0.2505 0.251 0.2515 0.252

(b)

0 10002000300040005000600070008000 9000 10000 0.248

0.2485 0.249 0.2495 0.25 0.2505 0.251 0.2515 0.252

(c)

0 100020003000 400050006000700080009000 10000 0.248

0.2485 0.249 0.2495 0.25 0.2505 0.251 0.2515 0.252

(d)

0 10002000300040005000 6000700080009000 10000 0.248

0.2485 0.249 0.2495 0.25 0.2505 0.251 0.2515 0.252

(e)

0 10002000300040005000600070008000 9000 10000 0.248

0.2485 0.249 0.2495 0.25 0.2505 0.251 0.2515 0.252

(f)

Figure 6. Graphs ofr_nfor the infinite Bernoulli convolutions in (1.1), where ρare selected Garsia numbers in Table 2.

0 200 400 600 800 1000 0.20

0.22 0.24 0.26 0.28 0.30

(a)

0 200 400 600 800 1000

0.20 0.22 0.24 0.26 0.28 0.30

(b)

0 200 400 600 800 1000

0.20 0.22 0.24 0.26 0.28 0.30

(c)

0 200 400 600 800 1000

0.20 0.22 0.24 0.26 0.28 0.30

(d)

0 200 400 600 800 1000

0.20 0.22 0.24 0.26 0.28 0.30

(e)

0 200 400 600 800 1000

0.20 0.22 0.24 0.26 0.28 0.30

(f)

Figure 7. Graphs ofrnfor the weighted infinite Bernoulli convolutions in (1.2) with weights w₁ = 1/4 and w₂ = 3/4, where ρ are the Garsia numbers in Table 2.

0 200 400 600 800 1000

0.40 0.45 0.50 0.55 0.60

(a)

0 200 400 600 800 1000

0.40 0.45 0.50 0.55 0.60

(b)

0 200 400 600 800 1000

0.40 0.45 0.50 0.55 0.60

(c)

0 200 400 600 800 1000

0.40 0.45 0.50 0.55 0.60

(d)

0 200 400 600 800 1000

0.40 0.45 0.50 0.55 0.60

(e)

0 200 400 600 800 1000

0.40 0.45 0.50 0.55 0.60

(f)

Figure 8. Graphs of An for the weighted infinite Bernoulli convolu- tions in (1.2) with weights w1 = 1/4 and w2 = 3/4, where ρ are the Garsia numbers in Table 2.